#497366

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,11)(8,13)(9,14,21,12)(10,15)(16,24)(17,23)(18,22,30,25)(19,27,32,26)(20,28)(29,34)(31,33,36,35);(1,2)(3,4)(5,7,10,16)(6,11,15,24)(8,17,20,29)(9,18,21,30)(12,25,14,22)(13,23,28,34)(19,31,32,36)(26,35,27,33);(6,12)(7,16)(9,21)(11,22)(13,26)(14,15)(17,29)(19,32)(23,33)(24,25)(27,28)(34,35);(2,4)(5,8,10,20)(6,13,15,28)(7,17,16,29)(9,19,21,32)(11,23,24,34)(12,26,14,27)(18,31,30,36)(22,33,25,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404562);